Next we bound the terms in (REF ). By applying a classical
inequality on unit-rate Poisson process (see Proposition
5.2 in [1]}), we have

P(0tT1(n)i|Ni,1(0t(n)ids)|>')  =P(0t(n)iT|Ni,1(t)|>'(n)i) 2e-(n)iTh('T),

where \(h\)  is a function defined by \(h(u)=(1+u)\log (1+u)-u\) .
Similarly,

 P(0tT1(n)i|Ni,2(0t(n)ii{ Y(n)i(s),n(n)im(n)i} ds)|>')
